Algebraic Number Theory

The origins of algebraic number theory.- I Algebraic Methods.- 1 Algebraic background.- 1.1 Rings and fields.- 1.2 Factorization of polynomials.- 1.3 Field extensions.- 1.4 Symmetric polynomials.- 1.5 Modules.- 1.6 Free abelian groups.- 2 Algebraic numbers.- 2.1 Algebraic numbers.- 2.2 Conjugates and discriminants.- 2.3 Algebraic integers.- 2.4 Integral bases.- 2.5 Norms and traces.- 3 Quadratic and cyclotomic fields.- 3.1 Quadratic fields.- 3.2 Cyclotomic fields.- 4 Factorization into irreducibles.- 4.1 Historical background.- 4.2 Trivial factorizations.- 4.3 Factorization into irreducibles.- 4.4 Examples of non-unique factorization into irreducibles.- 4.5 Prime factorization.- 4.6 Euclidean domains.- 4.7 Euclidean quadratic fields.- 4.8 Consequences of unique factorization.- 4.9 The Ramanujan-Nagell theorem.- 5 Ideals.- 5.1 Historical background.- 5.2 Prime factorization of ideals.- 5.3 The norm of an ideal.- II Geometric Methods.- 6 Lattices.- 6.1 Lattices.- 6.2 The quotient torus.- 7 Minkowski's theorem.- 7.1 Minkowski's theorem.- 7.2 The two-squares theorem.- 7.3 The four-squares theorem.- 8 Geometric representation of algebraic numbers.- 8.1 The space Lst.- 9 Class-group and class-number.- 9.1 The class-group.- 9.2 An existence theorem.- 9.3 Finiteness of the class-group.- 9.4 How to make an ideal principal.- 9.5 Unique factorization of elements in an extension ring.- III Number-Theoretic Applications.- 10 Computational methods.- 10.1 Factorization of a rational prime.- 10.2 Minkowski's constants.- 10.3 Some class-number calculations.- 10.4 Tables.- 11 Fermat's Last Theorem.- 11.1 Some history.- 11.2 Elementary considerations.- 11.3 Kummer's lemma.- 11.4 Kummer's Theorem.- 11.5 Regular primes.- 12 Dirichlet's Units Theorem.- 12.1 Introduction.- 12.2 Logarithmic space.- 12.3 Embedding the unit group in logarithmic space.- 12.4 Dirichlet's theorem.- Appendix 1 Quadratic Residues.- A.3 Quadratic Residues.- Appendix 2 Valuations.- References.